Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 9626725, 8 pages
http://dx.doi.org/10.1155/2016/9626725

Research Article

New Exponential Stability Results for Neutral Stochastic Systems with Distributed Delays

Jian Wang

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455002, China

Received 18 April 2016; Revised 12 September 2016; Accepted 9 October 2016

Academic Editor: Mohammad D. Aliyu

Copyright © 2016 Jian Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper studies the stability problem of a grey neutral stochastic system with distributed delays. The main feature of a grey system is that the parameters of system are evaluated by grey numbers, and system has the strong background in engineering applications. However, the literature dealing with the stability problem for grey system seems to be scarce. The main aim of this paper is to fill the gap. Based on the Lyapunov stability theorem and Itô’s formula, especially, using the decomposition technique of the continuous matrix-covered sets of grey matrix, we propose several novel sufficient conditions, which ensure our considered grey system in mean-square exponential stability and almost surely exponential stability. Furthermore, two examples are provided to show the effectiveness of the obtained results.

1. Introduction

In various physical and engineering systems, time-delay frequently causes oscillation, bifurcation, or instability. Meanwhile, the influence of stochastic phenomenon also cannot be ignored. So, time-delay stochastic system has gained many scholars’ attention and interest (see [1–9]). As pointed out in [10, 11], if the differences between nearby argument values are reduced, but the number of summands is incremental, we will find distributed delays phenomenon. In addition, we can describe many practical systems by neutral differential equations, Hence, neutral stochastic system with distributed delays has been extensively investigated by many scholars (see [12–20]). For instance, in [12], stability analysis of neutral type neural networks has been studied. Authors have proposed several delay-dependent criteria to ensure the networks to have a balance point. In [13], by the Lyapunov-Krasovkii functional, authors have studied the global exponential stability of neutral type neural networks with distributed time delays, and authors have presented new sufficient conditions of the system.

It is well known that we cannot obtain parameters of systems accurately. Naturally, in order to reflect many real systems, we have to evaluate the parameters of systems. When we evaluate the parameters of system by grey numbers, the system will be described uncertainly and turn into grey system (see [21]). Recently, grey system has come to play the important role in many fields and can be widely applied to physical and engineering systems, such as population ecology, communication systems, electric power systems, and so on. Due to the strong background in engineering applications, the stability analysis for grey system has become a focused topic of theoretical as well as practical importance. But, little attention has been taken to the stability problem for grey system except some paper [21–25]. In [21], the problem of exponential stability for a grey stochastic system with distributed delays has been studied, and the authors have proposed the delay-dependent criteria, which ensure the system in p-moment exponential robust stability. In [22], the authors have investigated robust stability problem for a grey stochastic nonlinear system with distributed delays, and several exponential stability criteria have been derived in terms of LMIs.

Motivated by the above discussion, using a similar method of [24], we will address the stability problem of grey system here. The main work and contribution of this paper are given as follows. (1) Compared with [24], this paper studies exponential stability problem of grey nonlinear systems with distributed delays for the first time, and the obtained results are more effective and general. (2) Different from traditional methods [10, 14, 20], for the uncertain systems, combining with concepts and conclusions of the Grey Theory (see Definitions 1 and 2 and Lemma 3), we adopt the new decomposition technique of the continuous matrix-covered sets of grey matrix. Then, several sufficient conditions are derived to ensure the systems in mean-square exponential stability and almost surely exponential stability. (3) Since the obtained criteria are presented LMIs, it can be easily solved by the Matlab LMI Control Toolbox. Meanwhile, two simple examples to illustrate the effectiveness of the obtained criteria are provided and solved.

The notation in this paper is fairly standard. The superscript “” represents the transpose and denotes the Euclidean norm. The notation , where is real symmetric matrix, means that is positive definite. Let be a probability space with a filtration and be the family of all continuous -valued functions on and be the family of all -measurable bounded -valued random variables .

2. Preliminaries and Problem Formulation

In this section, we consider the following grey system:

where , , and are interval matrices, and

are called the continuous matrix-covered sets of grey matrices , , and .

For grey system (1), we make the following assumptions and definitions:

(H1) is locally Lipschitz continuous and satisfies the linear growth condition.(H2)There are constants , , for arbitrary ; we have

Definition 1 (see [21]). Grey system (1) is mean square exponentially stable, if for all and arbitrary , , , there exist constants and , such that

Definition 2 (see [21]). Grey system (1) is almost surely exponentially robustly stable, if for all and arbitrary , , , we have

To prove our main results, the following lemmas are very important (in particular, Lemma 3).

Lemma 3 (see [21]). If is a grey matrix, is a number-covered sets of ; then for arbitrary matrix , we have

(1),(2),(3),

where , , , and .

Lemma 4 (see [26]). For any real vectors and any matrix with appropriate dimensions, , , we have

Lemma 5 (see [27]). For a given matrix with , , the following conditions are equivalent:

(1),(2), .

3. Main Results and Proofs

Several novel stability conditions are proposed to guarantee grey system (1) in the mean-square exponential stability and almost surely exponential stability. The main results are given in the following theorems and corollaries.

Theorem 6. System (1) is mean square exponentially stable, if and there exist symmetric matrices , , and scalars , , , , such that

where

Then, for all , we have

here,

and is the unique positive solution of the following equation

and , .

Proof. Using Lemma 5,

implies that

For convenience, let

Firstly, arbitrarily fix , and , , ; we introduce the following Lyapunov-Krasovskii functional:

Using Itô’s differential formula, we have

By applying Lemmas 3 and 4, it follows that

From assumptions (H2), we can get

Then, combining (18)–(27) and noting (7), we have

where is defined in (13).
Now, let , and by (28), we can get

Thus, it can be seen that

Set .
We have .
Since , and , then (12) has a unique solution .
By (30) and if (12) holds, we can obtain

On the other hand, it is easy to see

Then, it follows from (31) and (32) that

Note that

Then, we can derive

By (33), and note that definition of , it is easy to obtain

So, we also have

By (37) and noting , we can get

That is, (10) holds. This completes the proof of Theorem 6.

Corollary 7. System (1) is mean square exponentially stable, if and there exist symmetric matrices and scalars , , , , , such that

where

Then, for all , we have

here,

and is the unique positive solution of the following equation

and , .

Proof. Let , ; similar to the method used in Theorem 6, the proof of Corollary 7 is completed.

Remark 8. If the interval matrices , , and are replaced by deterministic matrices , , and , then grey system (1) will become the deterministic stochastic system:

Let , , and , and similar to the method used in Corollary 7, the criterion of the deterministic system (44) can be obtained.

Corollary 9. System (44) is mean square exponentially stable, if and there exist symmetric matrices and scalars , , , , , such that

where

Then, for all , we have

here,

and is the unique positive solution of the following equation

and , .

Theorem 10. Under the conditions of Theorem 6, grey system (1) is almost surely exponential stability. That is, for all , we have

where , , and satisfies (12).

Proof. Under the conditions of Theorem 6, system (1) is mean square exponentially stable, so we have

Using Doob’s martingale inequality, Cauchy inequality, from (51), we have

Then, using Borel-Cantelli lemma, and similar to the method used in Theorem (see [23, 24]), we complete the proof of Theorem 10.

4. Numerical Examples

Example 1. Consider a grey neutral systems with distributed delays:

Here, the lower bound and upper bound matrices of , , and are

Moreover,

Then, we have

By the calculation procedure programmed in [24], we can obtain and optimize scalars , , , and , which satisfy the criteria conditions in Theorem 6, and make the Lyapunov exponent much smaller. Now, the computed allowable values of constants and the Lyapunov exponent are shown in Table 1.
According to Theorem 6, and from Table 1, it is obvious that grey system (53) is exponentially stable in mean square, although the given system is described uncertainly. Moreover, it should be pointed out that [24] merely studies the exponential stability of a linear grey stochastic system. But, in this example, we can see that our work is suited to more general nonlinear grey stochastic systems. In the meantime, Theorem 6 only needs four positive scalars.

Table 1: Calculation values of , , , , and in Example 1.

Constants
The Lyapunov exponent

Values
0.5281
0.7163
0.8210
0.8516
1.1025

Example 2. Consider the stochastic system (44) with

For this deterministic system, if Corollary 9 in this paper is used, and by the Matlab LMI Control Toolbox, we can know that the Lyapunov exponent of this example is 1.0296.
For a comparison with the results of other methods and ours, the Lyapunov exponent is shown in Table 2.
It can be seen from Table 2 that,the Lyapunov exponent obtained by Corollary 9 is larger. That is, the result obtained by Corollary 9 is less conservative than that in [7, 15] and is an improvement of those results. Therefore, the criterion in Corollary 9 is valid. In addition, it is worth noting that since the parameters of given system are deterministic, the criterion in [7, 15] cannot be applied to Example 1. But Corollary 9 can be generalized, and we can obtain Corollary 7 to solve Example 1. Meanwhile, in this example, Corollary 9 only seeks a positive define matrix.

Table 2: Comparison of the Lyapunov exponent for different methods.

Methods
The Lyapunov exponent

Sun and Liu [7]
0.8985

Song et al. [15]
0.9962

Corollary 9
1.0296

5. Conclusion

This paper has focused on the exponential stability problem for a grey neutral stochastic system with distributed delays. By the Lyapunov stability theorem, Itô’s formula, and decomposition technique of the continuous matrix-covered sets of grey matrix, some novel stability criteria have been obtained, which ensure our grey system in the mean-square exponential stability and almost surely exponential stability. Finally, we provide two examples to show the effectiveness of the proposed method.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research reported here is supported by the National Natural Science Foundation of China (no. 11301009) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province (no. 16B110001).

References

E. Boukas and Z. Liu, Deterministic and Stochastic Time-Delay Systems, Birkhäuser, Boston, Mass, USA, 2002.

J. Cao and J. Wang, “Global exponential stability and periodicity of recurrent neural networks with time delays,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 5, pp. 920–931, 2005.

S. Arik, “Global robust stability analysis of neural networks with discrete time delays,” Chaos, Solitons and Fractals, vol. 26, no. 5, pp. 1407–1414, 2005.

M. S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, vol. 5, Dekker, New York, NY, USA, 2000.

X. Mao, “Exponential stability of stochastic delay interval systems with markovian switching,” IEEE Transactions on Automatic Control, vol. 47, no. 10, pp. 1604–1612, 2002.

S. Xu, J. Lam, and D. W. C. Ho, “Novel global robust stability criteria for interval neural networks with multiple time-varying delays,” Physics Letters A, vol. 342, no. 4, pp. 322–330, 2005.

J. Sun and G. P. Liu, “A new delay-dependent stability criterion for time-delay systems,” Asian Journal of Control, vol. 11, no. 4, pp. 427–431, 2009.

R. Z. Khas'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1981.

H. Huang, D. W. C. Ho, and J. Lam, “Stochastic stability analysis of fuzzy Hopfield neural networks with time-varying delays,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 52, no. 5, pp. 251–255, 2005.

S. Xu and T. Chen, “An LMI approach to the filter design for uncertain systems with distributed delays,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 51, no. 4, pp. 195–201, 2004.

Y. A. Fiagbedzi and A. E. Pearson, “A multistage reduction technique for feedback stabilizing distributed time-lag systems,” Automatica, vol. 23, no. 3, pp. 311–326, 1987.

J.-E. Feng, S. Y. Xu, and Y. Zou, “Delay-dependent stability of neutral type neural networks with distributed delays,” Neurocomputing, vol. 72, no. 10-12, pp. 2576–2580, 2009.

R. Rakkiyappan and P. Balasubramaniam, “New global exponential stability results for neutral type neural networks with distributed time delays,” Neurocomputing, vol. 71, no. 4, pp. 1039–1045, 2008.

W.-H. Chen and X. Lu, “Mean square exponential stability of uncertain stochastic delayed neural networks,” Physics Letters A, vol. 372, no. 7, pp. 1061–1069, 2008.

B. Song, J. H. Park, Z.-G. Wu, and Y. Zhang, “New results on delay-dependent stability analysis for neutral stochastic delay systems,” Journal of the Franklin Institute, vol. 350, no. 4, pp. 840–852, 2013.

J. H. Park and S. Won, “Stability of neutral delay-differential systems with nonlinear perturbations,” International Journal of Systems Science, vol. 31, no. 8, pp. 961–967, 2000.

R. Samli and S. Arik, “New results for global stability of a class of neutral-type neural systems with time delays,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 564–570, 2009.

S. S. Young, P. D. Scott, and N. M. Nasrabadi, “Object recognition using multilayer Hopfield neural network,” IEEE Transactions on Image Processing, vol. 6, no. 3, pp. 357–372, 1997.

L. Wan and J. Sun, “Mean square exponential stability of stochastic delayed Hopfield neural networks,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 343, no. 4, pp. 306–318, 2005.

Z. Wang, Y. Liu, and X. Liu, “On global asymptotic stability of neural networks with discrete and distributed delays,” Physics Letters A, vol. 345, no. 4–6, pp. 299–308, 2005.

C.-H. Su and S.-F. Liu, “The p-moment exponential robust stability for stochastic systems with distributed delays and interval parameters,” Applied Mathematics and Mechanics, vol. 30, no. 7, pp. 915–924, 2009.

C. H. Su and S. F. Liu, “Robust stability of grey stochastic nonlinear systems with distributed delays,” Mathematics in Practice and Theory, vol. 38, no. 22, pp. 218–223, 2008.

C. H. Su, “Robust stability of grey stochastic delay systems with impulsive effect,” Journal of Systems Science and Mathematical Sciences, vol. 32, no. 5, pp. 537–548, 2012.

C. H. Su and S. F. Liu, “Exponential robust stability of grey neutral stochastic systems with distributed delays,” Chinese Journal of Engineering Mathematics, vol. 27, no. 3, pp. 403–414, 2010.

J. Wang, “Exponential stability analysis for neutral stochastic systems with distributed delays,” IAENG International Journal of Applied Mathematics, vol. 45, no. 4, pp. 364–372, 2015.

A. Friedman, Stochastic Differential Equations and Their Applications, Academic Press, New York, NY, USA, 1976.

S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994.